The latter formula, known as Mehler's kernel (see [1]}), shows that
\(\,\mathrm {e}^{- \frac{s}{{E}_B}H_B}\,\in {S}_B\)  for all \(s>0\) .
The resolvent
\(R_{\lambda } := (H_B-\lambda {E}_B\mathbf {1})^{-1} ,\qquad \lambda -\frac{1}{2} \notin \mathbb {N}_0\) 
